In geometry, the tetraoctagonal tiling is a uniform tiling of the hyperbolic plane.
Tetraoctagonal tiling | |
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Poincaré disk model of the hyperbolic plane | |
Type | Hyperbolic uniform tiling |
Vertex configuration | (4.8)^{2} |
Schläfli symbol | r{8,4} or rr{8,8} rr(4,4,4) t_{0,1,2,3}(∞,4,∞,4) |
Wythoff symbol | 2 | 8 4 |
Coxeter diagram | or or |
Symmetry group | [8,4], (*842) [8,8], (*882) [(4,4,4)], (*444) [(∞,4,∞,4)], (*4242) |
Dual | Order-8-4 quasiregular rhombic tiling |
Properties | Vertex-transitive edge-transitive |
There are for uniform constructions of this tiling, three of them as constructed by mirror removal from the [8,4] or (*842) orbifold symmetry. Removing the mirror between the order 2 and 4 points, [8,4,1^{+}], gives [8,8], (*882). Removing the mirror between the order 2 and 8 points, [1^{+},8,4], gives [(4,4,4)], (*444). Removing both mirrors, [1^{+},8,4,1^{+}], leaves a rectangular fundamental domain, [(∞,4,∞,4)], (*4242).
Name | Tetra-octagonal tiling | Rhombi-octaoctagonal tiling | ||
---|---|---|---|---|
Image | ||||
Symmetry | [8,4] (*842) |
[8,8] = [8,4,1^{+}] (*882) = |
[(4,4,4)] = [1^{+},8,4] (*444) = |
[(∞,4,∞,4)] = [1^{+},8,4,1^{+}] (*4242) = or |
Schläfli | r{8,4} | rr{8,8} =r{8,4}^{1}/_{2} |
r(4,4,4) =r{4,8}^{1}/_{2} |
t_{0,1,2,3}(∞,4,∞,4) =r{8,4}^{1}/_{4} |
Coxeter | = | = | = or |
The dual tiling has face configuration V4.8.4.8, and represents the fundamental domains of a quadrilateral kaleidoscope, orbifold (*4242), shown here. Adding a 2-fold gyration point at the center of each rhombi defines a (2*42) orbifold.
In geometry, the rhombitetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of rr{8,4}. It can be seen as constructed as a rectified tetraoctagonal tiling, r{8,4}, as well as an expanded order-4 octagonal tiling or expanded order-8 square tiling.
Snub tetraoctagonal tilingIn geometry, the snub tetraoctagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of sr{8,4}.
Truncated tetraoctagonal tilingIn geometry, the truncated tetraoctagonal tiling is a semiregular tiling of the hyperbolic plane. There are one square, one octagon, and one hexakaidecagon on each vertex. It has Schläfli symbol of tr{8,4}.
*n42 symmetry mutations of quasiregular tilings: (4.n)^{2} | ||||||||
---|---|---|---|---|---|---|---|---|
Symmetry *4n2 [n,4] |
Spherical | Euclidean | Compact hyperbolic | Paracompact | Noncompact | |||
*342 [3,4] |
*442 [4,4] |
*542 [5,4] |
*642 [6,4] |
*742 [7,4] |
*842 [8,4]... |
*∞42 [∞,4] |
[ni,4] | |
Figures | ||||||||
Config. | (4.3)^{2} | (4.4)^{2} | (4.5)^{2} | (4.6)^{2} | (4.7)^{2} | (4.8)2 | (4.∞)^{2} | (4.ni)^{2} |
Dimensional family of quasiregular polyhedra and tilings: (8.n)^{2} | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Symmetry *8n2 [n,8] |
Hyperbolic... | Paracompact | Noncompact | ||||||||
*832 [3,8] |
*842 [4,8] |
*852 [5,8] |
*862 [6,8] |
*872 [7,8] |
*882 [8,8]... |
*∞82 [∞,8] |
[iπ/λ,8] | ||||
Coxeter | |||||||||||
Quasiregular figures configuration |
3.8.3.8 |
4.8.4.8 |
8.5.8.5 |
8.6.8.6 |
8.7.8.7 |
8.8.8.8 |
8.∞.8.∞ |
8.∞.8.∞ |
Uniform octagonal/square tilings | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
[8,4], (*842) (with [8,8] (*882), [(4,4,4)] (*444) , [∞,4,∞] (*4222) index 2 subsymmetries) (And [(∞,4,∞,4)] (*4242) index 4 subsymmetry) | |||||||||||
= = = |
= |
= = = |
= |
= = |
= |
||||||
{8,4} | t{8,4} |
r{8,4} | 2t{8,4}=t{4,8} | 2r{8,4}={4,8} | rr{8,4} | tr{8,4} | |||||
Uniform duals | |||||||||||
V8^{4} | V4.16.16 | V(4.8)^{2} | V8.8.8 | V4^{8} | V4.4.4.8 | V4.8.16 | |||||
Alternations | |||||||||||
[1^{+},8,4] (*444) |
[8^{+},4] (8*2) |
[8,1^{+},4] (*4222) |
[8,4^{+}] (4*4) |
[8,4,1^{+}] (*882) |
[(8,4,2^{+})] (2*42) |
[8,4]^{+} (842) | |||||
= |
= |
= |
= |
= |
= |
||||||
h{8,4} | s{8,4} | hr{8,4} | s{4,8} | h{4,8} | hrr{8,4} | sr{8,4} | |||||
Alternation duals | |||||||||||
V(4.4)^{4} | V3.(3.8)^{2} | V(4.4.4)^{2} | V(3.4)^{3} | V8^{8} | V4.4^{4} | V3.3.4.3.8 |
Uniform octaoctagonal tilings | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Symmetry: [8,8], (*882) | |||||||||||
= = |
= = |
= = |
= = |
= = |
= = |
= = | |||||
{8,8} | t{8,8} |
r{8,8} | 2t{8,8}=t{8,8} | 2r{8,8}={8,8} | rr{8,8} | tr{8,8} | |||||
Uniform duals | |||||||||||
V8^{8} | V8.16.16 | V8.8.8.8 | V8.16.16 | V8^{8} | V4.8.4.8 | V4.16.16 | |||||
Alternations | |||||||||||
[1^{+},8,8] (*884) |
[8^{+},8] (8*4) |
[8,1^{+},8] (*4242) |
[8,8^{+}] (8*4) |
[8,8,1^{+}] (*884) |
[(8,8,2^{+})] (2*44) |
[8,8]^{+} (882) | |||||
= | = | = | = = |
= = | |||||||
h{8,8} | s{8,8} | hr{8,8} | s{8,8} | h{8,8} | hrr{8,8} | sr{8,8} | |||||
Alternation duals | |||||||||||
V(4.8)^{8} | V3.4.3.8.3.8 | V(4.4)^{4} | V3.4.3.8.3.8 | V(4.8)^{8} | V4^{6} | V3.3.8.3.8 |
Uniform (4,4,4) tilings | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
Symmetry: [(4,4,4)], (*444) | [(4,4,4)]^{+} (444) |
[(1^{+},4,4,4)] (*4242) |
[(4^{+},4,4)] (4*22) | ||||||||
t_{0}(4,4,4) h{8,4} |
t0,1(4,4,4) h_{2}{8,4} |
t_{1}(4,4,4) {4,8}^{1}/_{2} |
t1,2(4,4,4) h_{2}{8,4} |
t_{2}(4,4,4) h{8,4} |
t0,2(4,4,4) r{4,8}^{1}/_{2} |
t_{0,1,2}(4,4,4) t{4,8}^{1}/_{2} |
s(4,4,4) s{4,8}^{1}/_{2} |
h(4,4,4) h{4,8}^{1}/_{2} |
hr(4,4,4) hr{4,8}^{1}/_{2} | ||
Uniform duals | |||||||||||
V(4.4)^{4} | V4.8.4.8 | V(4.4)^{4} | V4.8.4.8 | V(4.4)^{4} | V4.8.4.8 | V8.8.8 | V3.4.3.4.3.4 | V8^{8} | V(4,4)^{3} |
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